It is possible to solve a problem with continuous action spaces and no states with reinforcement learning?
A stateless RL problem can be reduced to a Multiarmed Bandit (MAB) problem. In such a scenario, taking an action will not change the state of the agent.

So, this is the setting of a conventional MAB problem: at each time step, the agent selects an action to either perform an exploration or exploitation move. It then records the reward of the taken action and updates its estimation/expectation of the usefulness of the action. Then, repeats the procedure (selection, observing, updating). 

To chose between exploration and exploitation moves, MAB agents adopt a strategy. The simplest one would probably be $\epsilon$-greedy which agent chooses the most rewarding actions most of the time (1-$\epsilon$ probability) or randomly selects an action ($\epsilon$ probability).